# On the regularity of D-modules generated by relative characters

## TL;DR

This paper introduces a new class of D-modules called K-admissible modules on homogeneous G-varieties, proving their regularity in certain spherical cases and exploring applications to relative characters, Harish-Chandra modules, and matrix coefficients.

## Contribution

It establishes the regularity of K-admissible D-modules on spherical varieties, extending the understanding of their holonomicity and growth properties in representation theory.

## Key findings

- K-admissible D-modules are regular holonomic on absolutely spherical varieties.
- Includes applications to relative characters, Harish-Chandra modules, and matrix coefficients.
- Provides growth estimates for K-admissible distributions using subanalytic geometry.

## Abstract

Following the ideas of Ginzburg, for a subgroup $K$ of a connected reductive $\mathbb{R}$-group $G$ we introduce the notion of $K$-admissible $D$-modules on a homogeneous $G$-variety $Z$. We show that $K$-admissible $D$-modules are regular holonomic when $K$ and $Z$ are absolutely spherical. This framework includes: (i) the relative characters attached to two spherical subgroups $H_1$ and $H_2$, provided that the twisting character $\chi_i$ factors through the maximal reductive quotient of $H_i$, for $i = 1, 2$; (ii) localization on $Z$ of Harish-Chandra modules; (iii) the generalized matrix coefficients when $K(\mathbb{R})$ is maximal compact. This complements the holonomicity proven by Aizenbud--Gourevitch--Minchenko. The use of regularity is illustrated by a crude estimate on the growth of $K$-admissible distributions which based on tools from subanalytic geometry.